Abstract
In this paper, we study polynomial identities with involution of an incidence algebra I(P, F) where P is a connected finite poset with an involution λ and F is a field of characteristic zero. At first, we also consider P of length at most 2 and then of length at most 3. Let and
denote, respectively, the λ-orthogonal and the λ-symplectic involutions of I(P, F). For the case that P has length at most 2 and
, we show that the
-identities and the
-identities of I(P, F) follow from the ordinary identity
. In that context, passing to the particular case
, where
is a poset called crown with 2n elements, and using the classification of the involutions on
, we show that, for all involutions ρ on
, every ρ-identity also follows from the ordinary identity
. For the case that P has length at most 3 and
, we determine the generators of the
-ideal
when every element of P that is neither minimal nor maximal is fixed by λ and, for such an element, there exists a unique minimal element of P that is comparable with it.
Communicated by Igor Klep
Acknowledgments
We thank the referee for the valuable suggestions and comments which improved this paper.
Disclosure statement
The authors report there are no competing interests to declare.